- #1
LAHLH
- 409
- 1
I'm trying to prove the conformal invariance (under [itex]g_{\mu\nu}\to\omega^2 g_{\mu\nu}[/itex]) of
[tex]
\bar{\Box}{\bar{\phi}}+\frac{1}{4}\frac{n-2}{n-1}\bar{R}\bar{\phi}
[/tex]
I've found that this equation is invariant upto a quantity proportional to
[tex]
g^{\mu\nu}\left[-\omega(\nabla_{\mu}\nabla_{\nu}\omega)\phi+(\omega^2-\omega)(\nabla_{\mu}\omega)(\nabla_{\nu} \phi)-\frac{n-4}{4}(\nabla_{\mu}\omega)(\nabla_{\nu} \phi)\right]
[/tex]
Here [itex]\bar{\phi}=\omega^{(2-n)/2}\phi[/itex], and the conformal transformations of other quantitities like box and the Ricci scalar are those found in Carroll Appendix.
How can I get rid of this extra unwanted quantity? (or have I simply made an algebraic error of some kind)
Thanks
[tex]
\bar{\Box}{\bar{\phi}}+\frac{1}{4}\frac{n-2}{n-1}\bar{R}\bar{\phi}
[/tex]
I've found that this equation is invariant upto a quantity proportional to
[tex]
g^{\mu\nu}\left[-\omega(\nabla_{\mu}\nabla_{\nu}\omega)\phi+(\omega^2-\omega)(\nabla_{\mu}\omega)(\nabla_{\nu} \phi)-\frac{n-4}{4}(\nabla_{\mu}\omega)(\nabla_{\nu} \phi)\right]
[/tex]
Here [itex]\bar{\phi}=\omega^{(2-n)/2}\phi[/itex], and the conformal transformations of other quantitities like box and the Ricci scalar are those found in Carroll Appendix.
How can I get rid of this extra unwanted quantity? (or have I simply made an algebraic error of some kind)
Thanks